I’m from Quebec and not familiar with the work as a whole.

To me the humor is more in the very informal phrasing of stuff like il lui a dit, le chef (which is rather unusual language for french fiction) and Paulin’s insistence that yes, he’s supposed to be afraid than in the precise formulation, although the fact it is a short, punchy sentence obviously helps.

I’m not a native speaker, but my French wife and I tried our best to raise our two kids to be bilingual in an English-speaking country and they both would often say just “il faut” instead of repeating the complete “il faut [avoir/faire/aller/whatever fits]”, so I agree with you that this might be how a kid would talk and that it does add humor to the exchange for me.
Regarding Nicolas’ correct use of “Si” to contradict the chef’s negative statement, I’m not sure if French kids raised in France struggle with mastering this contra-positive form of “yes” for false negative questions/statements, but ours (both in their 20s) still struggle with it and I would have found the exchange even funnier if Nicolas had responded with either “Oui, il faut!” or “Non, il faut!”

The trick here is in the use of “il faut” which indicates an obligation.

I think the closer tranlation in this context is :

“you should not be afraid;”
“yes, we should”,…”yes, we should”

Here the sentence “il ne faut pas avoir peur” shows the commun use of obligation verbs for an advice or a recommandation.

As a French native speaker, I would say that the use of “Si, il faut!” does not refer to a silly phrasing only used by kids. In this context it has a funny meaning because of the expected child’s intonation. Of course, it is a shortened form of answer and lacks the end of the sentence : “Si, il faut (avoir peur)!” and probably an adult would formulate this more often with the complete sentence. However, it is still normal everyday French to shortened sentences like this and to use “il faut” very often in everyday language.

“Si, il faut” is a little odd for two reasons. One reason is that the leader says “you don’t have to be afraid”, which like in English is an idiomatic way of saying “you shouldn’t be afraid”, the implied reasoning being that there is no reason to be afraid¹ and people should only be afraid if there is a sufficient reason to. The negation is that there is a reason to be afraid, and so it is admissible to be afraid; but the kid’s reply is that fright is not just permitted but mandatory: “yes, I/we must be afraid”.

The second reason is that the idiomatic way of saying that something is mandatory would be “il le faut”. Saying just “il faut” is in fact borderline ungrammatical: “il faut” requires an object.

The combination of a logically odd statement with an unusual phrasing both gives the impression that the child is panicking and gives a humorous overtone to the situation.

¹ _{Indeed, “there’s nothing to be afraid of” would be idiomatic and equivalent in English in this context, and so would “il n’y a pas de raison d’avoir peur” be in French.}